Introducing sequences In maths, we call a list of numbers in order a sequence. Each number in a sequence is called a term. 4, 8, 12, 16, 20, 24, 28, 32, ... 1st term 6th term If terms are next to each other they are referred to as consecutive terms. Pupils may ask the difference between a sequence of numbers and a number pattern. A sequence, unlike a number pattern, does not need to follow a rule or pattern. It can follow an irregular pattern, affected by different factors (e.g. the maximum temperature each day); or consist of a random set of numbers (e.g. numbers in the lottery draw). These ideas can be discussed in more detail during the plenary session at the end of the lesson. When we write out sequences, consecutive terms are usually separated by commas.

Infinite and finite sequences A sequence can be infinite. That means it continues forever. For example, the sequence of multiples of 10, 10, 20, 30, 40, 50, 60, 70, 80, 90 , ... is infinite. We show this by adding three dots at the end. If a sequence has a fixed number of terms it is called a finite sequence. For example, the sequence of two-digit square numbers 16, 25 ,36, 49, 64, 81 is finite.

Sequences and rules Some sequences follow a simple rule that is easy to describe. For example, this sequence 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, … continues by adding 3 each time. Each number in this sequence is one less than a multiple of three. Other sequences are completely random. For example, the sequence of winning raffle tickets in a prize draw. Ask pupils if they can see how the sequence 2, 5, 8, 11, 14, 17, … continues before revealing the solution. In maths we are mainly concerned with sequences of numbers that follow a rule.

Naming sequences Here are the names of some sequences which you may know already: 2, 4, 6, 8, 10, ... Even Numbers (or multiples of 2) 1, 3, 5, 7, 9, ... Odd numbers 3, 6, 9, 12, 15, ... Multiples of 3 5, 10, 15, 20, 25, ... Multiples of 5 This can be done as an oral activity. It may also be useful for pupils to copy these number patterns into their books. As each number sequence is revealed ask the name of the sequence before revealing it. Year 7 pupils should have met square numbers and triangular numbers in Year 6. They will be revisited in more detail later in the year. Pupils may describe the pattern of square numbers either as adding 3, adding 5, adding 7 etc. (i.e. adding consecutive odd numbers) or as 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5 etc. Pupils may need help verbalizing a rule to generate triangular numbers (i.e. add together consecutive whole numbers). 1, 4, 9, 16, 25, ... Square numbers 1, 3, 6, 10, 15, ... Triangular numbers

Ascending sequences +5 +5 +5 +5 +5 +5 +5 ×2 ×2 ×2 ×2 ×2 ×2 ×2 When each term in a sequence is bigger than the one before the sequence is called an ascending sequence. For example, The terms in this ascending sequence increase in equal steps by adding 5 each time. 2, 7, 12, 17, 22, 27, 32, 37, ... +5 +5 +5 +5 +5 +5 +5 The terms in this ascending sequence increase in unequal steps by starting at 0.1 and doubling each time. Stress the difference between sequences that increase in equal steps (linear sequences) and sequences that increase in unequal steps (non-linear) sequences. 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, ... ×2 ×2 ×2 ×2 ×2 ×2 ×2

Descending sequences –7 –7 –7 –7 –7 –7 –7 –1 –2 –3 –4 –5 –6 –7 When each term in a sequence is smaller than the one before the sequence is called a descending sequence. For example, The terms in this descending sequence decrease in equal steps by starting at 24 and subtracting 7 each time. 24, 17, 10, 3, –4, –11, –18, –25, ... –7 –7 –7 –7 –7 –7 –7 The terms in this descending sequence decrease in unequal steps by starting at 100 and subtracting 1, 2, 3, … Stress the difference between sequences that decrease in equal steps (linear sequences) and sequences that decrease in unequal steps (non-linear) sequences. 100, 99, 97, 94, 90, 85, 79, 72, ... –1 –2 –3 –4 –5 –6 –7

Sequences that decrease in equal steps Can you work out the next three terms in this sequence? 22, 16, 10, 4, –2, –8, –14, –20, ... –6 –6 –6 –6 –6 –6 –6 How did you work these out? This sequence starts with 22 and decreases by 6 each time. Introduce the word difference and encourage pupils to find the difference between consecutive terms. Remind pupils that they must check that every number in the sequence obeys the same rule. Each term in the sequence is two less than a multiple of 6. Sequences that increase or decrease in equal steps are called linear or arithmetic sequences.

Fibonacci-type sequences Can you work out the next three terms in this sequence? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... 1 + 1 1 + 2 2 + 3 3 + 5 5 + 8 8 + 13 13 + 21 21 + 34 How did you work these out? This sequence starts with 1, 1 and each term is found by adding together the two previous terms. Sequences of this type must be generated by two numbers. Ask pupils if this type of sequence can be descending. If the sequence is generated by two negative numbers then it will be a descending sequence. The Fibonacci sequence appears in many situations in nature. Ask pupils to research some examples on the Internet. This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it.

Describing and continuing sequences Here are some of the types of sequence you may come across: Sequences that increase or decrease in equal steps. These are called linear or arithmetic sequences. Sequences that increase or decrease in unequal steps by multiplying or dividing by a constant factor. These are called geometric sequences. Sequences that increase or decrease in unequal steps by adding or subtracting increasing or decreasing numbers. These are called quadratic sequences. Discuss each of these different types of sequences asking pupils to give examples for each one. The second type of sequence is a sequence of powers or geometric sequence, the third type is a quadratic sequence, and the fourth is a Fibonacci-type sequence. These can be broadly divided into two types: Sequences that increase or decrease in equal steps – linear sequences – and sequences that increase or decrease in unequal steps – non-linear sequences. Ask pupils how we can use the differences between consecutive terms to help us to recognize each type. In the first type of sequence the differences are constant. In the second type of sequence the differences form another geometric sequence. In the third type the differences form a linear sequence and so the second row of differences are constant. In the fourth type the differences form the same sequence. Sequences that increase or decrease by adding together the two previous terms. These are called Fibonacci sequences.

Types of Number

N Z Q R Natural Numbers 1, 2, 3, 4, 5, . . . Integers The counting numbers N Natural Numbers 1, 2, 3, 4, 5, . . . Include all the whole numbes and zero Z Integers . . . -3, -2, -1, 0, 1, 2, 3, . . . Include all the integers plus fractions Q Rational Numbers Include all the Rational Numbers plus numbers that cannot be written as fractions R Real Numbers

Factor Prime Number 2, 3, 5, 7, 11, 13, 17, . . . Prime Factor A factor of a number divides exactly into that number Factor eg: Factors of 14 are: 1, 2, 7 and 14 A number with exactly TWO factors: (1 and itself) Prime Number 2, 3, 5, 7, 11, 13, 17, . . . A factor of a number which is also a prime number is called a prime factor Prime Factor

A factor of a number which is also a prime number is called a prime factor

24 = 2 x 2 x 2 x 3 = 2 x 3 24 12 6 3 1 ÷ 2 ÷ 2 ÷ 2 ÷ 3 Prime Factor A factor of a number which is also a prime number is called a prime factor Prime Factor eg 1: Write 24 as a product of Prime Factors 24 12 6 3 1 Keep dividing by prime numbers until you get to an answer of 1 ÷ 2 ÷ 2 ÷ 2 ÷ 3 24 = 2 x 2 x 2 x 3 3 = 2 x 3

A factor of a number which is also a prime number is called a prime factor eg 2: Write 315 as a product of Prime Factors 315 105 35 7 1 Keep dividing by prime numbers until you get to an answer of 1 ÷ 3 ÷ 3 ÷ 5 ÷ 7 315 = 3 x 3 x 5 x 7 2 = 3 x 5 x 7

A factor of a number which is also a prime number is called a prime factor eg 1: Write 357 as a product of Prime Factors 357 119 17 1 Keep dividing by prime numbers until you get to an answer of 1 ÷ 3 ÷ 7 ÷ 17 357 = 3 x 7 x 17

Factors and Multiples

Definition of Factors and Multiples If one number is a factor of a second number or divides the second (as 3 is a factor of 12), then the second number is a multiple of the first (as 12 is a multiple of 3).

Components of Factors and Multiples Divisibility Tests Prime Numbers Greatest Common Factor Least Common Multiple

A quick way to see if numbers are divisible Divisibility Tests A quick way to see if numbers are divisible

Divisibility by 2 or 5 A number is divisible by 2 if the number represented by the units digit is divisible by 2. The number is divisible by 2 if the units digit is 0, 2, 4, 6, or 8. A number is divisible by 5 if the number represented by the units digit is divisible by 5. A number is divisible by 5 if its unit digit is 0 or 5.

Divisibility by 3 or 9 A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility by 4 or 6 A number is divisible if the number represented by the last two digits of a number is divisible by 4, then the original number will be divisible by 4. A number is divisible by both 2 or 3, then it is divisible by 6. If it is not divisible by 2 and 3, then it is not divisible by 6.

Prime Numbers

Definition of a Prime Number A number which is divisible by 1 and itself, only has 2 factors. The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Greatest Common Factor

Definition of Greatest Common Factor For any two nonzero whole numbers a and b, the greatest common factor, written GCF(a, b) is the greatest factor (divisor) of both a and b.

How to find Greatest Common Factor To find the GCF of 24 and 36: List all factors of 24: 1, 2, 3, 4, 6, 8, 12, and 24. List all factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors are 1, 2, 3, 4, 6, and 12. The greatest (largest) is 12, therefore 12 is the greatest common factor of 24 and 36.

Least Common Multiple

Definition of Least Common Multiple A number is called a common multiple of two numbers if it is a multiple of both. Common multiples of 5 and 7 are: 35, 70, 105, 140, 175, etc. Because 35 is the smallest common multiple it is known as the least common multiple. In other words, for any two nonzero whole numbers a and b, the least common multiple written LCM(a, b), is the smallest multiple of both a and b.

How to find Least Common Multiple For positive integers a and b, LCM(a,b): (a * b)/(GCF(a,b)) In other words, multiply the two numbers you are finding the least common multiple of and divide that answer by the greatest common factor of the two numbers. Also, when GCF(a, b) = 1, LCM(a, b) = a * b.

Find LCM (28, 44) 28 *44 = 1232 GCF (28, 44) = 4 1232/4 = 308 308 is the Least Common Multiple of 28 and 44.